/ l(x)dx o
added through all ages up to co, the upper limit of life. The expectation of remaining life ea for a person who has already reached age a will contain the same integral from a to GO.
Now if /JLX for one particular age x changes to fix + Ajux, and I(a)/I0 accordingly changes in the ratio 1 - A^, or in the absolute amount - [l(a)/lQ ] AHX , a > x, then the new integral for §0 can be seen to be the old one plus
CJ
-I
X
This is readily translated into 5-year age intervals. The effect on the expectation of life at birth of a rise A5MX in the death rate 5MX applying to ages x to x + 4 at last birthday will be approximated by
In words applied to our problem: A decrease in the death rate at age x will increase the expectation of life more the younger is x, both in proportion to the probability of living to that age 1X/1Q and in proportion to the expectation of subsequent life ex at that age. For United States females in 1966 the first factor, 1X[10, is about 7/8 at age 55 what it is at age 0, and the second factor, ex - 2H, is about 1/3 at age 55 what it is at age 0. For the two factors combined, an improvement in the age-specific rate at age 0 to 4 increases the expectation of e0 more than three times as much as a similar improvement at age 55 to 59.
But expectation of life is a notion oriented to individuals, and we are interested here in populations, and especially in their increase and decrease. The life table eQ can be translated into population terms as the number of individuals living at any one time, for each birth per year. In this model the births and deaths are the same, and the population total does not change. In such a stationary model the annual death rate d is the reciprocal of the expectation of life:percent drop in all mortality up to age 50, so that everyone born lives to that age, would increase long-run growth by about 4 percent. The situation was very different in the developing countries of 20 years ago, when the fall in mortality greatly increased the rate of growth and tended to make the population much younger.